Multifractal analysis for Bedford-McMullen carpets

Gap sequences and Topological properties of Bedford-McMullen sets

In this paper, we study the topological properties and the gap sequences of Bedford-McMullen sets. First, we introduce a topological condition, the component separation condition (CSC), and a geometric condition, the exponential rate condition (ERC). Then we prove that the CSC implies the ERC, and that both of them are sufficient conditions for obtaining the asymptotic estimate of gap sequences. We also explore topological properties of Bedford-McMullen sets and prove that all normal Bedford-McMullen sets with infinitely many connected components satisfy the CSC, from which we obtain the asymp…

无限维Bedford-McMullen海绵的平均维数理论

Mean dimension theory for infinite dimensional Bedford-McMullen sponges

本文针对$r\geq3$的Bedford-McMullen海绵系统建立平均维数理论，明确计算其度量平均维数与平均豪斯多夫维数，扩展了Tsukamoto的相关结果，并给出两者相等的特殊情形。

反射贝德-麦克穆伦地毯的豪斯多夫维度

Hausdorff dimension of reflected Bedford--McMullen carpets

研究反射贝德-麦克穆伦地毯的豪斯多夫维度，通过弱坐标投影熵控制维度，推导出显式公式并讨论反射模式对投影熵的影响。

Self embeddings of Bedford-McMullen carpets

Let $F \subseteq \mathbb{R}^2$ be a Bedford-McMullen carpet defined by multiplicatively independent exponents, and suppose that either $F$ is not a product set, or it is a product set with marginals of dimension strictly between $0$ and $1$. We prove that any similarity $g$ such that $g(F) \subseteq F$ is an isometry composed of reflections about lines parallel to the axes. Our approach utilizes the structure of tangent sets of $F$, obtained by "zooming in" on points of $F$, projection theorems for products of self-similar sets, and logarithmic commensurability type results for self similar se…

On the intermediate dimensions of Bedford-McMullen carpets

The intermediate dimensions of a set $Λ$, elsewhere denoted by $\dim_θΛ$, interpolates between its Hausdorff and box dimensions using the parameter $θ\in[0,1]$. Determining a precise formula for $\dim_θΛ$ is particularly challenging when $Λ$ is a Bedford-McMullen carpet with distinct Hausdorff and box dimension. In this direction, answering a question of Fraser, we show that $\dim_θΛ$ is strictly less than the box dimension of $Λ$ for every $θ<1$, moreover, the derivative of the upper bound is strictly positive at $θ=1$. We also improve on the lower bound obtained by Falconer, Fraser and Kempt…

相似变换下Bedford-McMullen地毯的正测度交集

Positive-measure intersections of Bedford-McMullen carpets under similitudes

本文证明Bedford-McMullen地毯在相似变换下若交集有正测度，则正交部分必为对角或反对角，且等距情形下仅差平移，加强了2010年猜想。

$\mathbb Z^2$ 中 Bedford-McMullen 型集合的维数

Dimensions of Bedford-McMullen type sets in $\mathbb Z^2$

本文研究 $\mathbb Z^2$ 中由扩张仿射映射生成的 Bedford-McMullen 型前向轨道的各种维数，给出显式公式，并揭示其与半群增长及对偶吸引子维数的关系。

On the dimensions of a family of overlapping self-affine carpets

We consider the dimensions of a family of self-affine sets related to the Bedford-McMullen carpets. In particular, we fix a Bedford-McMullen system and then randomise the translation vectors with the stipulation that the column structure is preserved. As such, we maintain one of the key features in the Bedford-McMullen set up in that alignment causes the dimensions to drop from the affinity dimension. We compute the Hausdorff, packing and box dimensions outside of a small set of exceptional translations, and also for some explicit translations even in the presence of overlapping. Our results r…

Bedford-McMullen 地毯的中间维数及其在 Lipschitz 等价性中的应用

Intermediate dimensions of Bedford-McMullen carpets with applications to Lipschitz equivalence

本文利用大偏差速率函数给出了 Bedford-McMullen 地毯中间维数的精确公式，揭示其可数多个相变及严格凹性，并证明中间维数相等等价于均匀 Bernoulli 测度的 Hausdorff 多重分形谱相等，从而为 Lipschitz 等价性提供必要条件。

The Hausdorff dimension of the projections of self-affine carpets

We study the orthogonal projections of a large class of self-affine carpets, which contains the carpets of Bedford and McMullen as special cases. Our main result is that if $Λ$ is such a carpet, and certain natural irrationality conditions hold, then every orthogonal projection of $Λ$ in a non-principal direction has Hausdorff dimension $\min(γ,1)$, where $γ$ is the Hausdorff dimension of $Λ$. This generalizes a recent result of Peres and Shmerkin on sums of Cantor sets.

Quasisymmetric geometry of the Julia sets of McMullen maps

We study the quasisymmetric geometry of the Julia sets of McMullen maps $f_λ(z)=z^m+λ/z^\ell$, where $\ell$, $m\geq 2$ are integers satisfying $1/\ell+1/m<1$ and $λ\in\mathbb{C}\setminus\{0\}$. If the free critical points of $f_λ$ are escaped to the infinity, we prove that the Julia set $J_λ$ of $f_λ$ is quasisymmetrically equivalent to either a standard Cantor set, a standard Cantor set of circles or a round Sierpiński carpet (which is also standard in some sense). If the free critical points are not escaped, we give a sufficient condition on $λ$ such that $J_λ$ is a Sierpiński carpet and pro…

Slicing Theorems and rigidity phenomena for self affine carpets

Let $F$ be a Bedford-McMullen carpet defined by independent exponents. We prove that $\overline{\dim}_B (\ell \cap F) \leq \max \lbrace \dim^* F -1,0 \rbrace$ for all lines $\ell$ not parallel to the principal axes, where $\dim^*$ is Furstenberg's star dimension (maximal dimension of a microset). We also prove several rigidity results for incommensurable Bedford-McMullen carpets, that is, carpets $F$ and $E$ such that all defining exponents are independent: Assuming various conditions, we find bounds on the dimension of the intersection of such carpets, show that self affine measures on them a…

某些扩张映射的紧不变集的维数

Dimensions of compact invariant sets of some expanding maps

研究了由整数值对角矩阵给出的环面上扩张非共形映射的紧不变集的豪斯多夫维数和满豪斯多夫维数测度. McMullen和Bedford发现了“一般Sierpinski地毯”的豪斯多夫维数，Kenyon和Peres证明了满豪斯多夫维数的测度在某些情况下是唯一的.我们推广这些结果，通过使用补偿函数来研究一般的Sierpinski地毯表示的移位的有限型。给出了一般Sierpinski地毯存在唯一的满豪斯多夫维数测度的条件，并研究了唯一测度的性质.

Invariance of multifractal spectrum of uniform self-affine measures and its applications

We study the bi-Lipschitz classification of Bedford-McMullen carpets which are totally disconnected. Let $E$ be a such carpet and let $μ_E$ be the uniform Bernoulli measure on $E$. We show that the multifractal spectrum and the doubling property of $μ_E$ are both invariant under a bi-Lipschitz map. Moreover, we show that if $μ_E$ and $μ_F$ are doubling, then a bi-Lipschitz map between $E$ and $F$ enjoys a certain measure preserving property.

高维Bedford-McMullen子移位的临界Hausdorff测度

Critical Hausdorff Measure for Higher-Dimensional Bedford-McMullen Subshifts

本文研究高维Bedford-McMullen子移位的临界Hausdorff测度，证明维数相等当且仅当临界测度为正且有限，并解决Peres猜想，推广至sofic子移位并构造两类反例。

Large slices through self affine carpets

Let $F\subseteq [0,1]^2$ be a Bedford-McMullen carpet defined by exponents $m>n$, that projects to $[0,1]$ on the $y$-axis. We show that under mild conditions on $F$, there are many non principle lines $\ell$ such that $\dim^* F\cap \ell = \dim^* F -1$, where $\dim^*$ is Furstenberg's star dimension (maximal dimension of a microset). This exhibits the sharpness of recent Furstenberg-type slicing theorems obtained by Algom (2020) about upper bounds on the dimension of every such slice.

Improved versions of some Furstenberg type slicing Theorems for self-affine carpets

Let $F$ be a Bedford-McMullen carpet defined by independent integer exponents. We prove that for every line $\ell \subseteq \mathbb{R}^2$ not parallel to the major axes, $$ \dim_H (\ell \cap F) \leq \max \left\lbrace 0,\, \frac{\dim_H F}{\dim^* F} \cdot (\dim^* F-1) \right\rbrace$$ and $$ \dim_P (\ell \cap F) \leq \max \left\lbrace 0,\, \frac{\dim_P F}{\dim^* F} \cdot (\dim^* F-1) \right\rbrace$$ where $\dim^*$ is Furstenberg's star dimension (maximal dimension of microsets). This improves the state of art results on Furstenberg type slicing Theorems for affine invariant carpets.

Dimensions of an overlapping generalization of Barański carpets

We determine the Hausdorff, packing and box-counting dimension of a family of self-affine sets generalizing Barański carpets. More specifically, we fix a Barański system and allow both vertical and horizontal random translations, while preserving the rows and columns structure. The alignment kept in the construction allows us to give expressions for these fractal dimensions outside of a small set of exceptional translations. Such formulas will coincide with those for the non-overlapping case, and thus provide examples where the box-counting and Hausdorff dimension do not necessarily agree. The…

Badly approximable points on self-affine sponges and the lower Assouad dimension

We highlight a connection between Diophantine approximation and the lower Assouad dimension by using information about the latter to show that the Hausdorff dimension of the set of badly approximable points that lie in certain non-conformal fractals, known as self-affine sponges, is bounded below by the dynamical dimension of these fractals. In particular, for self-affine sponges with equal Hausdorff and dynamical dimensions, the set of badly approximable points has full Hausdorff dimension in the sponge. Our results, which are the first to advance beyond the conformal setting, encompass both …

On the convergence rate of the chaos game

This paper studies how long it takes the orbit of the chaos game to reach a certain density inside the attractor of a strictly contracting iterated function system of which we only assume that its lower dimension is positive. We show that the rate of growth of this cover time is determined by the Minkowski dimension of the push-forward of the shift invariant measure with exponential decay of correlations driving the chaos game. Moreover, we bound the expected value of the cover time from above and below with multiplicative logarithmic correction terms. As an application, for Bedford-McMullen c…

Hausdorff dimension of certain random self--affine fractals

In this work we are interested in the self--affine fractals studied by Gatzouras and Lalley and by the author which generalize the famous general Sierpinski carpets studied by Bedford and McMullen. We give a formula for the Hausdorff dimension of sets which are randomly generated using a finite number of self-affine transformations each one generating a fractal set as mentioned before, with some technical hypotheses. The choice of the transformation is random according to a Bernoulli measure. The formula is given in terms of the variational principle for the dimension.

Mean Hausdorff dimension of some infinite dimensional fractals

Mean Hausdorff dimension is a dynamical version of Hausdorff dimension. It provides a way to dynamicalize geometric measure theory. We pick up the following three classical results of fractal geometry. (1) The calculation of Hausdorff dimension of homogeneous sets in the circle. (2) The coincidence of Hausdorff and Minkowski dimensions for self-similar sets. (3) The calculation of Hausdorff dimension of Bedford--McMullen carpets. We develop their analogues for mean Hausdorff dimension: (1) The calculation of mean Hausdorff dimension of homogeneous systems in the infinite dimensional torus. (2)…

Dimension of self-affine sets with holes

In this paper we compute the dimension of a class of dynamically defined non-conformal sets. Let $X\subseteq\mathbb{T}^2$ denote a Bedford-McMullen set and $T:X\to X$ the natural expanding toral endomorphism which leaves $X$ invariant. For an open set $U\subset X$ we let X_U={x\in X : T^k(x)\not\in U \text{for all}k}. We investigate the box and Hausdorff dimensions of $X_U$ for both a fixed Markov hole and also when $U$ is a shrinking metric ball. We show that the box dimension is controlled by the escape rate of the measure of maximal entropy through $U$, while the Hausdorff dimension depends…

Local structure of self-affine sets

The structure of a self-similar set with open set condition does not change under magnification. For self-affine sets the situation is completely different. We consider planar self-affine Cantor sets E of the type studied by Bedford, McMullen, Gatzouras and Lalley, for which the projection onto the horizontal axis is an interval. We show that within small square neighborhoods of almost each point x in E, with respect to many product measures on address space, E is well approximated by product sets of an interval and a Cantor set. Even though E is totally disconnected, the limit sets have the p…

随机自仿贝德福德-麦克马伦地毯的Assouad谱

The Assouad spectrum of random self-affine carpets

本文研究随机自仿贝德福德-麦克马伦地毯，推导其几乎必然Assouad谱和拟Assouad维数，发现非共形情形下它们通常不同于盒维数或Assouad维数，并给出示例说明差异。

Assouad type dimensions for self-affine sponges

We study the Assouad and lower dimensions of self-affine sponges; the higher dimensional analogue of the planar self-affine carpets of Bedford and McMullen. Our techniques involve the weak tangents of Mackay and Tyson as well as regularity properties of doubling measures in the context studied by Bylund and Gudayol.

关于某些仿射不变集的豪斯多夫维数和盒维数相等性

On the coincidence of the Hausdorff and box dimensions for some affine-invariant sets

研究了某些仿射不变集的豪斯多夫维数和盒维数相等性，证明了在s≤2时三者等价，且在s≥3时发现新的现象，通过概率方法建立了贝福德-麦克穆伦海绵体的等价性。

Scaling scenery of $(\times m,\times n)$ invariant measures

We study the scaling scenery and limit geometry of invariant measures for the non-conformal toral endomorphism $(x,y) \mapsto (mx \mod 1,ny \mod 1)$ that are Bernoulli measures for the natural Markov partition. We show that the statistics of the scaling can be described by an ergodic CP-chain in the sense of Furstenberg. Invoking the machinery of CP-chains yields a projection theorem for Bernoulli measures, which generalises in part earlier results by Hochman-Shmerkin and Ferguson-Jordan-Shmerkin. We also give an ergodic theoretic criterion for the dimension part of Falconer's distance set con…

Assouad type spectra for some fractal families

In a previous paper we introduced a new `dimension spectrum', motivated by the Assouad dimension, designed to give precise information about the scaling structure and homogeneity of a metric space. In this paper we compute the spectrum explicitly for a range of well-studied fractal sets, including: the self-affine carpets of Bedford and McMullen, self-similar and self-conformal sets with overlaps, Mandelbrot percolation, and Moran constructions. We find that the spectrum behaves differently for each of these models and can take on a rich variety of forms. We also consider some applications, in…

sofic自仿射分形的改进维数理论

Improved dimension theory of sofic self-affine fractals

该研究建立了任意欧氏空间中自仿射分形Hausdorff维数的组合表达式，推广了平面sofic集相关研究，给出高维sofic集精确维数及平面两类维数相等的条件，同时补充了相关先行研究说明与后续工作进展。